Theorem elrnmpt2d 5956

Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
elrnmpt2d.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmpt2d.2	⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Assertion

Ref	Expression
elrnmpt2d	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem elrnmpt2d

Step	Hyp	Ref	Expression
1		elrnmpt2d.2	. 2 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹)
2		elrnmpt2d.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	elrnmpt 5948	. . 3 ⊢ (𝐶 ∈ ran 𝐹 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
4	3	ibi 270	. 2 ⊢ (𝐶 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
5	1, 4	syl 18	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ↦ cmpt 5192  ran crn 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is used by:  ablsimpg1gend  20181  elrgspnlem4  33574  elrgspnsubrunlem1  33576